No, they are not the same thing and both come in various sizes and configurations... The edger is generally a hand held tool but they do make a few wheel (push) types, while the trencher is a bigger tool generally on wheels and possibly the smallest models might be hand held but I think it would be more like a hand held tiller (not like a trimmer hand held). The edger is generally for more shallow trenches, the trencher should dig deeper, the depth of the trench is limited to each machine, specifically.
I guess you could say the largest edgers are where the smallest trenchers take over, if I have it right, I think I do.
Also depends on the blades, some edgers are single blade and can dig a hairline trench but hold it at an angle and it digs wider, however for a slightly wider trench it would be best and easiest to get a multi-blade tool in the first place, see what I'm saying.
And no, I don't think you can reasonably put a wire down a single-blade hairline trench in the dirt, I'd pick a tool that will dig 4-6 inches deep and at least 1 but 2-3 inches wide, not too much wider either.
I don't think this is no 60-100 foot project, you're likely going to encircle a yard and then you'll have extra footage to deal with where the driveway is, and what have you, probably you're looking at covering several thousand feet. But I do not know, you might wish to either measure or get some idea of how long this thing needs to be, it will make a difference because I can tell you I would not want to dig a 4-thousand foot trench with my single blade hand held edger :laugh:
So you will want to likely check into renting a tool, my recommendation would be to ask some of those guys, also keep in mind how DEEP you need your trench and how LONG (because wear and tear and the fact you want to get this done in under 4 hours could become an issue).
Probably either a large multi-blade wheel unit edger, or a small unit (wheels) trench digger, what I'm thinking you'll need...
Just my opinion.